**Some Theorems in General Topology**

- Every closed interval in
**R**of finite length is compact. More is true: In**R**`n`, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem). - Every continuous image of a compact space is compact.
- Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
- A compact subset of a Hausdorff space is closed.
- Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in
**R**is connected. - Every compact finite-dimensional manifold can be embedded in some Euclidean space
**R**`n`. - The continuous image of a connected space is connected.
- Every metric space is paracompact and Hausdorff, and thus normal.
- The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
- The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
- Any open subspace of a Baire space is itself a Baire space.
- The Baire category theorem: If
*X*is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. - On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
- Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.

General topology also has some surprising connections to other areas of mathematics. For example:

- In number theory, Fürstenberg's proof of the infinitude of primes.

See also some counter-intuitive theorems, e.g. the Banach–Tarski paradox.

Read more about this topic: Topology, Topology Topics

### Famous quotes containing the word general:

“Anti-Nebraska, Know-Nothings, and *general* disgust with the powers that be, have carried this county [Hamilton County, Ohio] by between seven and eight thousand majority! How people do hate Catholics, and what a happiness it was to show it in what seemed a lawful and patriotic manner.”

—Rutherford Birchard Hayes (1822–1893)